Consider a plasma in which, in the local mean rest frame of the electrons, the electron stress tensor has the form (20.35) with e_z the direction of the magnetic field. The following analysis for the electrons can be carried out independently for the ions, resulting in the same formulas.

(a) Show that

where (v²_||)  is the mean-square electron velocity parallel to B, and (|v_⊥|²) is the mean-square velocity orthogonal to B. (The velocity distributions are not assumed to be Maxwellian.)

(b) Consider a fluid element with length l along the magnetic field and cross sectional area A orthogonal to the field. Let v̅ be the mean velocity of the electrons (v̅ = 0 in the mean electron rest frame), and let θ and σ_jk be the expansion and shear of the mean electron motion as computed from v̅ (Sec. 13.7.1). Show that

where b = B/|B| = e_z is a unit vector in the direction of the magnetic field.

(c) Assume that the timescales for compression and shearing are short compared with those for Coulomb scattering and anomalous scattering: τ ≪ t_{D, e}. Show, using the laws of energy and particle conservation, that

(d) Show that

(e) Show that, when the fluid is expanding or compressing entirely perpendicular to B, with no expansion or compression along B, the pressures change in accord with the adiabatic indices of Eq. (20.36a). Show, similarly, that when the fluid expands or compresses along B, with no expansion or compression in the perpendicular direction, the pressures change in accord with the adiabatic indices of Eq. (20.36b).

(f) Hence derive the so-called double adiabatic or CGL equations of state:

valid for changes on timescales long compared with the cyclotron period but short compared with all Coulomb collision and anomalous scattering times.

Equations 20.36.

Transcribed Image Text:

Pe|| = nm₂(v²), Pe₁ = _ _nme(|v₁|³), 2 (20.42)